How to store a password as securely in Chrome Extension?

This is a lightning-rod issue. See http://blog.elliottkember.com/chromes-insane-password-security-strategy for more. The position most consistent with Chrome's would be to encourage your users to use whole-disk encryption and to lock their screen when away from a logged-in machine. It's difficult for userland code like an extension (or a browser, for that matter) to properly implement secure storage, where "properly" means "resistant to a password-recovery utility that anyone can download from the internet."

You should file a feature request. It might be possible to expose a system-level API that does provide similar security to the underlying OS's keychain.